What happens when I use REM?#

The workflow of Readout-Error Mitigation (REM) in Mitiq is represented in the figure below.


Workflow of the REM technique in Mitiq.#

  • The user provides a QPROGRAM, (i.e. a quantum circuit defined via any of the supported frontends)).

  • Mitiq leaves the input circuit unmodified.

  • The unmodified circuit is executed via a user-defined Executor.

  • REM modifies the measurement results by transforming by an inverse confusion matrix.

  • The error mitigated expectation value is returned to the user.

With respect to the workflows of other error-mitigation techniques (e.g. ZNE or PEC), REM does not require the modification of the circuit. The results of the execution of the single unmodified circuit are what get modified. For this reason, the circuit generation step is trivial and the work of the technique happens in the final inference step.

As shown in How do I use REM?, the function execute_with_rem() applies REM behind the scenes and directly returns the error-mitigated expectation value. In the next sections instead, we show how one can apply REM at a lower level, i.e., by by applying each step independently:

  • Obtaining measurement results directly from the executor;

  • Mitigating the measurements with an inverse confusion matrix

Obtaining measurements results directly from the executor#

Let’s use the same example from How do I use REM?:

from cirq import LineQubit, Circuit, X, measure_each

qreg = [LineQubit(i) for i in range(2)]
circuit = Circuit(X.on_each(*qreg), measure_each(*qreg))
0: ───X───M───

1: ───X───M───

Let’s create a noisy readout executor to have erroneous values to work with:

from functools import partial

import numpy as np
from cirq.experiments.single_qubit_readout_calibration_test import (

from mitiq import MeasurementResult

def noisy_readout_executor(
    circuit, p0: float = 0.01, p1: float = 0.01, shots: int = 8192
) -> MeasurementResult:
    # Replace with code based on your frontend and backend.
    simulator = NoisySingleQubitReadoutSampler(p0, p1)
    result = simulator.run(circuit, repetitions=shots)

    return MeasurementResult(
            # q[2:-1] is necessary to convert "q(number)" into "number"
            for k in result.measurements.keys()
            for q in k.split(",")

# Use a noisy executor that has a 25% chance of flipping
p_flip = 0.25
noisy_executor = partial(noisy_readout_executor, p0=p_flip, p1=p_flip)

With our noisy executor we can obtain the raw measurement results with Executor.run():

from mitiq.executor.executor import Executor

executor = Executor(noisy_executor)
result = executor.run([circuit])
noisy_result = result[0]
{'11': 4713, '00': 460, '01': 1479, '10': 1540}

Here we can see that instead of only getting '11' results we get other bitstrings. We’ll see how to correct these in the next section.

Mitigating the measurements with an inverse confusion matrix#

For this example we used a simple, uncorrelated readout error model where each qubit has a separate probability of flipping from \(|0\rangle \rightarrow |1\rangle\) and \(|1\rangle \rightarrow |0\rangle\) during readout. More details of constructing this inverse confusion matrix are presented in What additional options are available when using REM?.

from mitiq.rem import generate_inverse_confusion_matrix

inverse_confusion_matrix = generate_inverse_confusion_matrix(2, p_flip, p_flip)
[[ 2.25 -0.75 -0.75  0.25]
 [-0.75  2.25  0.25 -0.75]
 [-0.75  0.25  2.25 -0.75]
 [ 0.25 -0.75 -0.75  2.25]]

Next, we utilize our noisy results and the inverse confusion matrix in order to obtain mitigated results:

from mitiq.rem.inverse_confusion_matrix import mitigate_measurements

mitigated_result = mitigate_measurements(noisy_result, inverse_confusion_matrix)
{'11': 8192}

Now, we can see more of the '11' results that we would expect. These mitigated results would subsequently be used by any later stages that compute expectation values of observables.

The details of how the measurements are mitigated are presented in What is the theory behind REM?.